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Mirrors > Home > ILE Home > Th. List > supisoti | Unicode version |
Description: Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.) |
Ref | Expression |
---|---|
supiso.1 | |
supiso.2 | |
supisoex.3 | |
supisoti.ti |
Ref | Expression |
---|---|
supisoti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supisoti.ti | . . . . . . 7 | |
2 | 1 | ralrimivva 2491 | . . . . . 6 |
3 | supiso.1 | . . . . . . 7 | |
4 | isoti 6862 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | 2, 5 | mpbid 146 | . . . . 5 |
7 | 6 | r19.21bi 2497 | . . . 4 |
8 | 7 | r19.21bi 2497 | . . 3 |
9 | 8 | anasss 396 | . 2 |
10 | isof1o 5676 | . . . 4 | |
11 | f1of 5335 | . . . 4 | |
12 | 3, 10, 11 | 3syl 17 | . . 3 |
13 | supisoex.3 | . . . 4 | |
14 | 1, 13 | supclti 6853 | . . 3 |
15 | 12, 14 | ffvelrnd 5524 | . 2 |
16 | 1, 13 | supubti 6854 | . . . . . 6 |
17 | 16 | ralrimiv 2481 | . . . . 5 |
18 | 1, 13 | suplubti 6855 | . . . . . . 7 |
19 | 18 | expd 256 | . . . . . 6 |
20 | 19 | ralrimiv 2481 | . . . . 5 |
21 | supiso.2 | . . . . . . 7 | |
22 | 3, 21 | supisolem 6863 | . . . . . 6 |
23 | 14, 22 | mpdan 417 | . . . . 5 |
24 | 17, 20, 23 | mpbi2and 912 | . . . 4 |
25 | 24 | simpld 111 | . . 3 |
26 | 25 | r19.21bi 2497 | . 2 |
27 | 24 | simprd 113 | . . . 4 |
28 | 27 | r19.21bi 2497 | . . 3 |
29 | 28 | impr 376 | . 2 |
30 | 9, 15, 26, 29 | eqsuptid 6852 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 wrex 2394 wss 3041 class class class wbr 3899 cima 4512 wf 5089 wf1o 5092 cfv 5093 wiso 5094 csup 6837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-isom 5102 df-riota 5698 df-sup 6839 |
This theorem is referenced by: infisoti 6887 infrenegsupex 9357 infxrnegsupex 11000 |
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